Numerical Methods for Ordinary Differential Equations

540 Pages·2016·5.62 MB·English

by  J. C. Butcher

Numerical Methods for
Ordinary Differential
Equations
--------- THIRD EDITION - - - - -
J.C. Butcher
WILEY
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Numerical Methods for Ordinary
Differential Equations
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TrimSize:152mmx229mm Butcher f01.tex V1-05/31/2016 10:41A.M. Pageiii
Numerical Methods for Ordinary
Differential Equations
J. C. Butcher
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TrimSize:152mmx229mm Butcher f01.tex V1-05/31/2016 10:41A.M. Pageiv
Thiseditionfirstpublished2016
©2016,JohnWiley&Sons,Ltd
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Contents
Foreword xiii
Prefacetothefirstedition xv
Prefacetothesecondedition xix
Prefacetothethirdedition xxi
1 DifferentialandDifferenceEquations 1
10 DifferentialEquationProblems 1
100 Introductiontodifferentialequations 1
101 TheKeplerproblem 4
102 Aproblemarisingfromthemethodoflines 7
103 Thesimplependulum 11
104 Achemicalkineticsproblem 14
105 TheVanderPolequationandlimitcycles 16
106 TheLotka–Volterraproblemandperiodicorbits 18
107 TheEulerequationsofrigidbodyrotation 20
11 DifferentialEquationTheory 22
110 Existenceanduniquenessofsolutions 22
111 Linearsystemsofdifferentialequations 24
112 Stiffdifferentialequations 26
12 FurtherEvolutionaryProblems 28
120 Many-bodygravitationalproblems 28
121 Delayproblemsanddiscontinuoussolutions 30
122 Problemsevolvingonasphere 33
123 FurtherHamiltonianproblems 35
124 Furtherdifferential-algebraicproblems 36
13 DifferenceEquationProblems 38
130 Introductiontodifferenceequations 38
131 Alinearproblem 39
132 TheFibonaccidifferenceequation 40
133 Threequadraticproblems 40
134 Iterativesolutionsofapolynomialequation 41
135 Thearithmetic-geometricmean 43
vi NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS
14 DifferenceEquationTheory 44
140 Lineardifferenceequations 44
141 Constantcoefficients 45
142 Powersofmatrices 46
15 LocationofPolynomialZeros 50
150 Introduction 50
151 Lefthalf-planeresults 50
152 Unitdiscresults 52
Concludingremarks 53
2 NumericalDifferentialEquationMethods 55
20 TheEulerMethod 55
200 IntroductiontotheEulermethod 55
201 Somenumericalexperiments 58
202 Calculationswithstepsizecontrol 61
203 Calculationswithmildlystiffproblems 65
204 CalculationswiththeimplicitEulermethod 68
21 AnalysisoftheEulerMethod 70
210 FormulationoftheEulermethod 70
211 Localtruncationerror 71
212 Globaltruncationerror 72
213 ConvergenceoftheEulermethod 73
214 Orderofconvergence 74
215 Asymptoticerrorformula 78
216 Stabilitycharacteristics 79
217 Localtruncationerrorestimation 84
218 Roundingerror 85
22 GeneralizationsoftheEulerMethod 90
220 Introduction 90
221 Morecomputationsinastep 90
222 Greaterdependenceonpreviousvalues 92
223 Useofhigherderivatives 92
224 Multistep–multistage–multiderivativemethods 94
225 Implicitmethods 95
226 Localerrorestimates 96
23 Runge–KuttaMethods 97
230 Historicalintroduction 97
231 Secondordermethods 98
232 Thecoefficienttableau 98
233 Thirdordermethods 99
234 Introductiontoorderconditions 100
235 Fourthordermethods 101
236 Higherorders 103
237 ImplicitRunge–Kuttamethods 103
238 Stabilitycharacteristics 104
239 Numericalexamples 108
CONTENTS vii
24 LinearMultistepMethods 111
240 Historicalintroduction 111
241 Adamsmethods 111
242 Generalformoflinearmultistepmethods 113
243 Consistency,stabilityandconvergence 113
244 Predictor–correctorAdamsmethods 115
245 TheMilnedevice 117
246 Startingmethods 118
247 Numericalexamples 119
25 TaylorSeriesMethods 120
250 IntroductiontoTaylorseriesmethods 120
251 Manipulationofpowerseries 121
252 AnexampleofaTaylorseriessolution 122
253 Othermethodsusinghigherderivatives 123
254 Theuseoffderivatives 126
255 Furthernumericalexamples 126
26 MultivalueMulitistageMethods 128
260 Historicalintroduction 128
261 PseudoRunge–Kuttamethods 128
262 Two-stepRunge–Kuttamethods 129
263 Generalizedlinearmultistepmethods 130
264 Generallinearmethods 131
265 Numericalexamples 133
27 IntroductiontoImplementation 135
270 Choiceofmethod 135
271 Variablestepsize 136
272 Interpolation 138
273 ExperimentswiththeKeplerproblem 138
274 Experimentswithadiscontinuousproblem 139
Concludingremarks 142
3 Runge–KuttaMethods 143
30 Preliminaries 143
300 Treesandrootedtrees 143
301 Trees,forestsandnotationsfortrees 146
302 Centralityandcentres 147
303 Enumerationoftreesandunrootedtrees 150
304 Functionsontrees 153
305 Somecombinatorialquestions 155
306 Labelledtreesanddirectedgraphs 156
307 Differentiation 159
308 Taylor’stheorem 161
viii NUMERICALMETHODSFORORDINARYDIFFERENTIALEQUATIONS
31 OrderConditions 163
310 Elementarydifferentials 163
311 TheTaylorexpansionoftheexactsolution 166
312 Elementaryweights 168
313 TheTaylorexpansionoftheapproximatesolution 171
314 Independenceoftheelementarydifferentials 174
315 Conditionsfororder 174
316 Orderconditionsforscalarproblems 175
317 Independenceofelementaryweights 178
318 Localtruncationerror 180
319 Globaltruncationerror 181
32 LowOrderExplicitMethods 185
320 Methodsoforderslessthan4 185
321 Simplifyingassumptions 186
322 Methodsoforder4 189
323 Newmethodsfromold 195
324 Orderbarriers 200
325 Methodsoforder5 204
326 Methodsoforder6 206
327 Methodsofordergreaterthan6 209
33 Runge–KuttaMethodswithErrorEstimates 211
330 Introduction 211
331 Richardsonerrorestimates 211
332 Methodswithbuilt-inestimates 214
333 Aclassoferror-estimatingmethods 215
334 ThemethodsofFehlberg 221
335 ThemethodsofVerner 223
336 ThemethodsofDormandandPrince 223
34 ImplicitRunge–KuttaMethods 226
340 Introduction 226
341 Solvabilityofimplicitequations 227
342 MethodsbasedonGaussianquadrature 228
343 Reflectedmethods 233
344 MethodsbasedonRadauandLobattoquadrature 236
35 StabilityofImplicitRunge–KuttaMethods 243
350 A-stability,A(α)-stabilityandL-stability 243
351 CriteriaforA-stability 244
352 Pade´approximationstotheexponentialfunction 245
353 A-stabilityofGaussandrelatedmethods 252
354 Orderstars 253
355 OrderarrowsandtheEhlebarrier 256
356 AN-stability 259
357 Non-linearstability 262
358 BN-stabilityofcollocationmethods 265
359 TheV andW transformations 267
CONTENTS ix
36 ImplementableImplicitRunge–KuttaMethods 272
360 ImplementationofimplicitRunge–Kuttamethods 272
361 DiagonallyimplicitRunge–Kuttamethods 273
362 Theimportanceofhighstageorder 274
363 Singlyimplicitmethods 278
364 Generalizationsofsinglyimplicitmethods 283
365 EffectiveorderandDESIREmethods 285
37 ImplementationIssues 288
370 Introduction 288
371 Optimalsequences 288
372 Acceptanceandrejectionofsteps 290
373 Errorperstepversuserrorperunitstep 291
374 Control-theoreticconsiderations 292
375 Solvingtheimplicitequations 293
38 AlgebraicPropertiesofRunge–KuttaMethods 296
380 Motivation 296
381 EquivalenceclassesofRunge–Kuttamethods 297
382 ThegroupofRunge–Kuttatableaux 299
383 TheRunge–Kuttagroup 302
384 Ahomomorphismbetweentwogroups 308
385 AgeneralizationofG1 309
386 SomespecialelementsofG 311
387 Somesubgroupsandquotientgroups 314
388 Analgebraicinterpretationofeffectiveorder 316
39 SymplecticRunge–KuttaMethods 323
390 Maintainingquadraticinvariants 323
391 Hamiltonianmechanicsandsymplecticmaps 324
392 Applicationstovariationalproblems 325
393 Examplesofsymplecticmethods 326
394 Orderconditions 327
395 Experimentswithsymplecticmethods 328
Concludingremarks 331
4 LinearMultistepMethods 333
40 Preliminaries 333
400 Fundamentals 333
401 Startingmethods 334
402 Convergence 335
403 Stability 336
404 Consistency 336
405 Necessityofconditionsforconvergence 338
406 Sufficiencyofconditionsforconvergence 339
41 TheOrderofLinearMultistepMethods 344
410 Criteriafororder 344
411 Derivationofmethods 346
412 Backwarddifferencemethods 347